YES(O(1),O(n^2)) 175.41/60.07 YES(O(1),O(n^2)) 175.41/60.07 175.41/60.07 We are left with following problem, upon which TcT provides the 175.41/60.07 certificate YES(O(1),O(n^2)). 175.41/60.07 175.41/60.07 Strict Trs: 175.41/60.07 { rev(ls) -> r1(ls, empty()) 175.41/60.07 , r1(empty(), a) -> a 175.41/60.07 , r1(cons(x, k), a) -> r1(k, cons(x, a)) } 175.41/60.07 Obligation: 175.41/60.07 derivational complexity 175.41/60.07 Answer: 175.41/60.07 YES(O(1),O(n^2)) 175.41/60.07 175.41/60.07 We use the processor 'matrix interpretation of dimension 1' to 175.41/60.07 orient following rules strictly. 175.41/60.07 175.41/60.07 Trs: { r1(empty(), a) -> a } 175.41/60.07 175.41/60.07 The induced complexity on above rules (modulo remaining rules) is 175.41/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 175.41/60.07 component(s). 175.41/60.07 175.41/60.07 Sub-proof: 175.41/60.07 ---------- 175.41/60.07 TcT has computed the following triangular matrix interpretation. 175.41/60.07 175.41/60.07 [rev](x1) = [1] x1 + [2] 175.41/60.07 175.41/60.07 [r1](x1, x2) = [1] x1 + [1] x2 + [0] 175.41/60.07 175.41/60.07 [empty] = [2] 175.41/60.07 175.41/60.07 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 175.41/60.07 175.41/60.07 The order satisfies the following ordering constraints: 175.41/60.07 175.41/60.07 [rev(ls)] = [1] ls + [2] 175.41/60.07 >= [1] ls + [2] 175.41/60.07 = [r1(ls, empty())] 175.41/60.07 175.41/60.07 [r1(empty(), a)] = [1] a + [2] 175.41/60.07 > [1] a + [0] 175.41/60.07 = [a] 175.41/60.07 175.41/60.07 [r1(cons(x, k), a)] = [1] a + [1] x + [1] k + [0] 175.41/60.07 >= [1] a + [1] x + [1] k + [0] 175.41/60.07 = [r1(k, cons(x, a))] 175.41/60.07 175.41/60.07 175.41/60.07 We return to the main proof. 175.41/60.07 175.41/60.07 We are left with following problem, upon which TcT provides the 175.41/60.07 certificate YES(O(1),O(n^2)). 175.41/60.07 175.41/60.07 Strict Trs: 175.41/60.07 { rev(ls) -> r1(ls, empty()) 175.41/60.07 , r1(cons(x, k), a) -> r1(k, cons(x, a)) } 175.41/60.07 Weak Trs: { r1(empty(), a) -> a } 175.41/60.07 Obligation: 175.41/60.07 derivational complexity 175.41/60.07 Answer: 175.41/60.07 YES(O(1),O(n^2)) 175.41/60.07 175.41/60.07 We use the processor 'matrix interpretation of dimension 1' to 175.41/60.07 orient following rules strictly. 175.41/60.07 175.41/60.07 Trs: { rev(ls) -> r1(ls, empty()) } 175.41/60.07 175.41/60.07 The induced complexity on above rules (modulo remaining rules) is 175.41/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 175.41/60.07 component(s). 175.41/60.07 175.41/60.07 Sub-proof: 175.41/60.07 ---------- 175.41/60.07 TcT has computed the following triangular matrix interpretation. 175.41/60.07 175.41/60.07 [rev](x1) = [1] x1 + [2] 175.41/60.07 175.41/60.07 [r1](x1, x2) = [1] x1 + [1] x2 + [0] 175.41/60.07 175.41/60.07 [empty] = [1] 175.41/60.07 175.41/60.07 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 175.41/60.07 175.41/60.07 The order satisfies the following ordering constraints: 175.41/60.07 175.41/60.07 [rev(ls)] = [1] ls + [2] 175.41/60.07 > [1] ls + [1] 175.41/60.07 = [r1(ls, empty())] 175.41/60.07 175.41/60.07 [r1(empty(), a)] = [1] a + [1] 175.41/60.07 > [1] a + [0] 175.41/60.07 = [a] 175.41/60.07 175.41/60.07 [r1(cons(x, k), a)] = [1] a + [1] x + [1] k + [0] 175.41/60.07 >= [1] a + [1] x + [1] k + [0] 175.41/60.07 = [r1(k, cons(x, a))] 175.41/60.07 175.41/60.07 175.41/60.07 We return to the main proof. 175.41/60.07 175.41/60.07 We are left with following problem, upon which TcT provides the 175.41/60.07 certificate YES(O(1),O(n^2)). 175.41/60.07 175.41/60.07 Strict Trs: { r1(cons(x, k), a) -> r1(k, cons(x, a)) } 175.41/60.07 Weak Trs: 175.41/60.07 { rev(ls) -> r1(ls, empty()) 175.41/60.07 , r1(empty(), a) -> a } 175.41/60.07 Obligation: 175.41/60.07 derivational complexity 175.41/60.07 Answer: 175.41/60.07 YES(O(1),O(n^2)) 175.41/60.07 175.41/60.07 We use the processor 'matrix interpretation of dimension 2' to 175.41/60.07 orient following rules strictly. 175.41/60.07 175.41/60.07 Trs: { r1(cons(x, k), a) -> r1(k, cons(x, a)) } 175.41/60.07 175.41/60.07 The induced complexity on above rules (modulo remaining rules) is 175.41/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 175.41/60.07 component(s). 175.41/60.07 175.41/60.07 Sub-proof: 175.41/60.07 ---------- 175.41/60.07 TcT has computed the following triangular matrix interpretation. 175.41/60.07 175.41/60.07 [rev](x1) = [1 2] x1 + [2] 175.41/60.07 [0 1] [2] 175.41/60.07 175.41/60.07 [r1](x1, x2) = [1 2] x1 + [1 0] x2 + [0] 175.41/60.07 [0 1] [0 1] [2] 175.41/60.07 175.41/60.07 [empty] = [0] 175.41/60.07 [0] 175.41/60.07 175.41/60.07 [cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 175.41/60.07 [0 0] [0 1] [2] 175.41/60.07 175.41/60.07 The order satisfies the following ordering constraints: 175.41/60.07 175.41/60.07 [rev(ls)] = [1 2] ls + [2] 175.41/60.07 [0 1] [2] 175.41/60.07 > [1 2] ls + [0] 175.41/60.07 [0 1] [2] 175.41/60.07 = [r1(ls, empty())] 175.41/60.07 175.41/60.07 [r1(empty(), a)] = [1 0] a + [0] 175.41/60.07 [0 1] [2] 175.41/60.07 >= [1 0] a + [0] 175.41/60.07 [0 1] [0] 175.41/60.07 = [a] 175.41/60.07 175.41/60.07 [r1(cons(x, k), a)] = [1 0] a + [1 0] x + [1 2] k + [4] 175.41/60.07 [0 1] [0 0] [0 1] [4] 175.41/60.07 > [1 0] a + [1 0] x + [1 2] k + [0] 175.41/60.07 [0 1] [0 0] [0 1] [4] 175.41/60.07 = [r1(k, cons(x, a))] 175.41/60.07 175.41/60.07 175.41/60.07 We return to the main proof. 175.41/60.07 175.41/60.07 We are left with following problem, upon which TcT provides the 175.41/60.07 certificate YES(O(1),O(1)). 175.41/60.07 175.41/60.07 Weak Trs: 175.41/60.07 { rev(ls) -> r1(ls, empty()) 175.41/60.07 , r1(empty(), a) -> a 175.41/60.07 , r1(cons(x, k), a) -> r1(k, cons(x, a)) } 175.41/60.07 Obligation: 175.41/60.07 derivational complexity 175.41/60.07 Answer: 175.41/60.07 YES(O(1),O(1)) 175.41/60.07 175.41/60.07 Empty rules are trivially bounded 175.41/60.07 175.41/60.07 Hurray, we answered YES(O(1),O(n^2)) 175.41/60.07 EOF